From simple to complex networks: inherent structures, barriers and valleys in the context of spin glasses

Z. Burda, A. Krzywicki ¹, O. C. Martin ², Z. Tabor

Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 73 (2006) 036110

Given discrete degrees of freedom (spins) on a graph interacting via an energy function, what can be said about the energy local minima and associated inherent structures? Using the lid algorithm in the context of a spin glass energy function, we investigate the properties of the energy landscape for a variety of graph topologies. First, we find that the multiplicity of the inherent structures generically has a lognormal distribution. Furthermore, the quenched and annealed values of the corresponding entropy are different except for the Sherrington-Kirkpatrick model. Second, we find simple scaling laws for the growth of the height of the energy barrier between the two degenerate ground states and the size of the associated valleys. For finite connectivity models, changing the topology of the underlying graph does not modify qualitatively the energy landscape, but at the quantitative level the models can differ substantially.

• 1. Laboratoire de Physique Théorique d’Orsay (LPT),
  CNRS : UMR8627 – Université Paris XI – Paris Sud
• 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
  CNRS : UMR8626 – Université Paris XI – Paris Sud